LLG
Advanced Math and Science Pilot Class
Mathematics, Grade 10
2014 – 2015
Paris – Abu Dhabi
Chapter 12 : Usual Functions
Exercise 5 : Give the best window frame to study on your GDC the graph of the
following functions :
(a) 𝑓(𝑥) = 𝑥 2 − 4𝑥 + 3
(b) 𝑔(𝑥) = −2𝑥 2 + 3𝑥 + 1
(c) ℎ(𝑥) = 0.15 𝑥 2 − 3𝑥 − 1.25
Exercise 1 :
(a) Determine the expression of the linear function such that : 𝑓(−2) =
1 and 𝑓(6) = 5. Draw the graph of 𝑓. Give the variations of 𝑓 and justify.
(b) Determine the expression of the linear function so that the image of −1 equals 4
and the image of 3 equals −4.
(c) Determine the expression of the linear function such that : 𝑓(702) = 237 and
𝑓(−297) = −96.
Exercise 2 : We consider the function defined on [−6 ; 7] by :
1
2
Exercise 6 : 𝑓 is the function defined by 𝑓(𝑥) = (𝑥 − 2)2 − 1. Justifying the steps,
give a double inequality for 𝑓(𝑥) in each case :
(a) 𝑥 ∈ [2 ; 3]
Exercise 7 : Give the variations of the following functions, and their turning point :
(a) 𝑓(𝑥) = 2(𝑥 − 4)2 + 1
1
2
(b) ℎ(𝑥) = (𝑥 + 5)2 +
−𝑥 + 3
𝑓𝑜𝑟 𝑥 ∈ [−6 ; −2[
2𝑥 + 9
𝑓𝑜𝑟 𝑥 ∈ [−2 ; 1[
𝑓(𝑥) =
5
𝑓𝑜𝑟 𝑥 ∈ [1 ; 4[
𝑓𝑜𝑟 𝑥 ∈ [4 ; 7]
{−2𝑥 + 10
(b) 𝑥 ∈ [−3 ; −1]
1
3
(c) 𝑔(𝑥) = −3(𝑥 + 1)2 − 8
(d) 𝜑(𝑥) = −4(𝑥 − 5)2 − 7
Exercise 8 : We consider the function 𝑓: 𝑥 → −2𝑥 2 − 12𝑥 + 2
(a) Expand −2(𝑥 + 3)2 + 20.
(b) Show that for all 𝑥 ∈ ℝ, 𝑓(𝑥) ≤ 20.
(a) Calculate 𝑓(−5), 𝑓(−2), 𝑓(0), 𝑓(2), 𝑓(4), 𝑓(5), 𝑓(7).
(b) Sketch the curve of this function in an orthonormal frame of the plane.
(c) Prove that 𝑓 is decreasing on [−3 ; +∞[.
(d) Without any more justification, draw the table of variations of 𝑓.
Exercise 3 : What can you say about 𝑥 2 when :
(a)
(b)
(c)
(d)
(e)
(e) Evaluate 𝑓(0) and 𝑓(−6).
𝑥≥3
𝑥 < −5
−2 ≤ 𝑥 ≤ 1
−7 ≤ 𝑥 ≤ 0
− √3 < 𝑥 < −√2
(f) Deduce the solutions of 𝑓(𝑥) ≥ 2 from the previous questions.
Exercise 9 : Let 𝑓 be a function defined on ℝ by 𝑓(𝑥) = 𝑎 (𝑥 − 𝛼)2 + 𝛽 with 𝑎, 𝛼, 𝛽
real numbers, 𝑎 non null. We name 𝐶𝑓 its graph in an orthonormal frame of the
plane. Find the express of the function, knowing that :
Exercise 4 :
* this curve is symmetrical about the straight line with equation 𝑥 = −3,
(a) Compare : (10−10 + 2000)2 and (10−10 − 2000)2.
(b) Arrange in ascending order the squares of the numbers :
* the function has a maximum in −4
9
−2.05, −3, 5
5
, 2.
* 𝐴(0 ; −13) ∈ 𝐶𝑓
Exercise 10 : We consider three parabolas and their three equations. Match the
correct equation to each curve.
(d) Without any more justification, draw the table of variations of 𝑓.
(e) Factorize 𝑓 using its standard form.
(f) Solve −2(𝑥 − 3)(𝑥 + 1) > 0 and deduce the solutions of 𝑓(𝑥) > 0. Justify.
Exercise 12 : Let 𝑓 be the square function 𝑓(𝑥) = 𝑥 2 and 𝑔 the linear function with
𝑓(𝑥) = 0.5 𝑥 2 + 𝑥 + 1
𝑔(𝑥) = 𝑥(𝑥 − 3)
ℎ(𝑥) = 3 − (𝑥 − 1)2
graph 𝒟 pasing through the points 𝐴(1; 0) and 𝐵(3 ; 8).
(a) Determine the expression of 𝑔(𝑥).
(b) Using your GDC, conjecture the position of 𝒟 with respect to 𝐶𝑓 .
(c) Prove algebraically that those two lines have only one point in common.
Find its coordinates.
(d) Justify the position of the two curves.
Exercise 13 :
(a) Compare the reciprocal of 2 − √5 and 2 − √3.
9
5
(b) Arrange in ascending order the reciprocals of the numbers : −2.05, −3, ,
5
2
Exercise 14 : Are the following statements true or false ? Justify.
1
1
(a) If 𝑥 > 1 then 𝑥 < 1.
(b) If 𝑥 < 1 then 𝑥 > 1.
Exercise 15 : Find the domain of the following functions :
𝑓(𝑥) =
2𝑥 − 8
4𝑥 + 5
𝑔(𝑥) =
−2𝑥 + 3
−2𝑥
ℎ(𝑥) =
𝑥
0.2𝑥 + 5
𝑘(𝑥) =
4𝑥 + 3
−0.5𝑥 − 6
Exercise 16 :
(a) Show that for all 𝑥 ≠ −3 we have
Exercise 11 : We consider the function 𝑓: 𝑥 → −2𝑥 2 + 4𝑥 + 6
(a) Show the steps that lead you to find the standard form of 𝑓 (Without using
it, show that it is −2(𝑥 − 1)2 + 8) ).
(b) Prove that for all 𝑥 ∈ ℝ, 𝑓(𝑥) ≤ 8.
(c) Prove that 𝑓 is increasing on ] − ∞ ; 1].
2𝑥−1
𝑥+3
7
= 2 − 𝑥+3.
(b) Use the previous result to give the variations of 𝑓 on ]−3 ; +∞[.
Exercise 17 : Give the variations of the following functions on the given interval :
5
𝑓(𝑥) = 4 − −𝑥+2 𝐼 = ]−∞ ; 2[
4
𝑔(𝑥) = 3 + 7−2𝑥
7
𝐼 = ]2 ; +∞[